Mathematics > Logic

Title:On principles between $Σ_1$- and $Σ_2$-induction, and monotone enumerations

Abstract: We show that many principles of first-order arithmetic, previously only known
to lie strictly between $\Sigma_1$-induction and $\Sigma_2$-induction, are
equivalent to the well-foundedness of $\omega^\omega$.
Among these principles are the iteration of partial functions ($P\Sigma_1$)
of Hájek and Paris, the bounded monotone enumerations principle
(non-iterated, BME$_1$) by Chong, Slaman, and Yang, the relativized
Paris-Harrington principle for pairs, and the totality of the relativized
Ackermann-Péter function.
With this we show that the well-foundedness of $\omega^\omega$ is a far more
widespread than usually suspected.
Further, we investigate the $k$-iterated version of the bounded monotone
iterations principle (BME$_k$), and show that it is equivalent to the
well-foundedness of the $k+1$-height $\omega$-tower.